# Definition:Bounded Below Mapping/Real-Valued

< Definition:Bounded Below Mapping(Redirected from Definition:Bounded Below Real-Valued Function)

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*This page is about real-valued functions which are bounded below. For other uses, see Definition:Bounded Below.*

## Contents

## Definition

Let $f: S \to \R$ be a real-valued function.

Then $f$ is **bounded below on $S$** by the lower bound $L$ if and only if:

- $\forall x \in S: L \le \map f x$

That is, if and only if the set $\set {\map f x: x \in S}$ is bounded below in $\R$ by $L$.

## Unbounded Below

Let $f: S \to \R$ be a real-valued function.

Then $f$ is **unbounded below on $S$** if and only if it is not bounded below on $S$:

- $\neg \exists L \in \R: \forall x \in S: L \le \map f x$

## Also see

## Sources

- 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 3$: Bounds of a Function - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 7.13$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**bound**:**1.**(of a function)